REVIEW 3 major objections 5 minor 33 references
Comparing the three W-like states with the W state
T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash
Pith's one-line read The paper claims that the three W-like states |ϑ′⟩, |η′⟩, and |ξ′⟩ each possess one maximally mixed qubit and one maximal one-versus-two tangle, leave two-qubit states that are more entangled than the W state's after tracing out a qubit, an
desk verdict A correct but modest paper: the explicit entanglement calculations and protocols check out, but the novelty is overstated and two key proofs are left to 'a complicated calculation.' read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery is the Schmidt decomposition of W-SLOCC states, |ψ⟩ = λ0|000⟩ + λ1 e^{iϕ}|100⟩ + λ2|101⟩ + λ3|110⟩, together with closed-form expressions for four LU-invariant entanglement measures (von Neumann entanglement entropy, 2-tangles, negativities, and one-versus-two tangles) in terms of the λ coefficients. The paper uses these formulas to identify, for each choice of distinguished qubit, the family of states with maximal marginal entropy, solves for the unique member of that family that also maximizes all average measures, and verifies the resource protocols by exhibiting orthogonal four-qubit bases for teleportation and superdense coding.
What would settle it
Directly compute the 2-tangle and negativity for a W-class state with the phase term present (λ1 > 0), for example |ψ⟩ = (1/2)|000⟩ + (1/10)|100⟩ + λ2|101⟩ + λ3|110⟩, and check whether the claimed maximal values in Lemmas 1–3 still occur only when λ1 = 0; alternatively, numerically maximize the average negativity over the family in Eq. (22) and test whether a state with λ0, λ2 not both equal to 1/2 beats (√2+1)/12.
Extended reading notes
Core claim
The central discovery is that within the SLOCC W class there exist locally-unitarily inequivalent states whose single-qubit marginal can be maximally mixed and whose bipartition entanglement can be maximal — properties previously associated mainly with the GHZ class. Concretely, |ϑ⟩ = (1/√2)|000⟩ + (1/2)|101⟩ + (1/2)|110⟩ (and two cyclic variants |η⟩, |ξ⟩) realize S(ρ_A) = ln 2 and τ_{A(BC)} = 1. The tables show that after tracing out any one of the other two qubits, the 2-tangle and negativity of the remaining pair exceed those of the W state. The author also proves, by explicit construction of four orthogonal measurement bases, that all six states (the three original forms and their Schmid
Load-bearing premise
The paper assumes without independent derivation that its formulas for von Neumann entropy, 2-tangles, negativities, and one-versus-two tangles, which it imports from earlier work, correctly describe every state in the W SLOCC class; if any formula is only conditionally valid, the maxima, the comparison tables, and the protocol conclusions would lose their support.
Editorial extensions
If this is right
- If the results are correct, these six states provide a concrete advantage over the W state for any task sensitive to single-particle loss: dropping one qubit still leaves a more entangled pair.
- The states join the GHZ state as resources for perfect teleportation and superdense coding, but they belong to the W SLOCC class, which may be easier to prepare or more robust in practical settings.
- The pattern shows that maximal mixedness of one qubit is not exclusive to GHZ; a one-parameter family within the W class can achieve it while retaining full three-partite entanglement.
- The explicit protocols in the appendices give ready-to-use measurement and correction sets for implementing two-qubit teleportation and dense coding with these states.
Reading between the lines
- The results suggest a trade-off principle: as one qubit approaches maximal mixedness, the other two qubits become more entangled, so the three states may be optimal for distributed tasks where one party holds a distinguished qubit.
- One could experimentally test the predicted robustness by preparing |ϑ⟩, sending it through a lossy or noisy channel, and comparing the 2-tangle of the surviving pair against the W state prepared under the same conditions.
- The same optimization approach could be extended to search W-class states that maximize other entanglement monotones, or generalized to four-qubit W-like states, though the uniqueness proofs would need to be checked numerically.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies three W-like three-qubit states |ϑ′⟩, |η′⟩, |ξ′⟩ and their Schmidt forms |ϑ⟩, |η⟩, |ξ⟩, comparing them with the W state using four entanglement measures: the 2-tangles τxy, negativities, von Neumann entanglement entropy, and the 1-vs-2 tangles τx(yz). It claims that each of the three states has one single-qubit reduced density matrix with maximal entropy ln 2 and one tangle equal to 1, that tracing out a suitable qubit leaves a two-qubit state more entangled than the corresponding reduced state of the W state, and that these states enable perfect teleportation and superdense coding while the W state does not. Theorems 1–3 assert uniqueness of the maximizers within restricted Schmidt families; Results 1–4 compare W-state averages; Theorems 4–5 give explicit protocols.
Significance. If the results are correct, the paper provides a useful, small catalogue of W-class states with asymmetric entanglement extremality and explicit teleportation/dense-coding circuits. The explicit calculations I checked—the Schmidt forms, Tables 1–4, Lemma 1, Theorem 1, and the protocols for |ξ⟩ and |ϑ⟩—are internally consistent and reproducible from the displayed formulas. The contribution is incremental but publishable in a specialized quantum-information venue, provided the proof gaps identified below are filled. The main weakness is that several central assertions are supported only by 'a calculation yields' or by external citations rather than by derivations in the manuscript.
major comments (3)
- [Theorems 2 and 3 (Sections 'THE STATES HAVING THE MAXIMAL VNEE S(ρB)' and 'S(ρC)')] The proofs of Theorems 2 and 3 consist solely of 'A complicated calculation yields' with no displayed equations or derivative bounds. These theorems are the main uniqueness results for |η⟩ and |ξ⟩. Since |η⟩ and |ξ⟩ are qubit permutations of |ϑ⟩, the derivative analysis used in Theorem 1 and Properties 1.1–1.4 can be adapted; please supply the actual calculation or a symmetrization argument.
- [Result 1 (Section 'A_neg, Aτx(yz), Avnee, AND Aτxy FOR THE W STATE')] The maximization of A_neg over the W-like states in Eq. (2) is asserted via 'A calculation shows' and 'One can verify'. This extremum is load-bearing for the W-state comparison. Please include an explicit two-variable optimization (or a bounding inequality) showing that the conditional maximum (√5−1)/6 occurs at λ0=λ2=λ3=1/√3.
- [Theorems 4 and 5 (Sections on teleportation and superdense coding)] The claim that the W state cannot be used for perfect teleportation or superdense coding is not proved in this manuscript; it rests entirely on references [20,21,24]. Since this negative statement is part of the paper's headline comparison, please state explicitly the no-go criterion used in those references (e.g., the W-class condition for deterministic perfect teleportation) and verify that the standard W state violates it. Without this, the comparative claim is not self-contained.
minor comments (5)
- [Lemma 1 proof, Eq. (19)] The sentence 'To guarantee λ0 ≥ 0, λ1 in Eq. (19) must vanish' should be 'To guarantee a real solution for λ0'; non-negativity is not the obstruction—reality of λ0 is.
- [Appendix A, teleportation for |ϑ⟩] In the classical communication mapping after the |ϑ⟩ expansion, '00 to |ς+⟩_{a12}' should read '|ς+⟩_{a23}', since the measurement is on qubits a,2,3. Please check all such subscripts in the appendices.
- [Terminology, Preliminary and Summary] The paper first defines W-like states as γ|001⟩+β|010⟩+α|100⟩, but later also calls the Schmidt forms |ϑ⟩,|η⟩,|ξ⟩ 'W-like states'. These families overlap only after local unitaries; please clarify the terminology to avoid confusion.
- [Result 3] Result 3 is proved by citing [6] without stating the corresponding theorem. Please quote the relevant result or give the argument, so the reader can verify the claim without consulting the earlier paper.
- [Tables 1–4 and text] Fractions such as √5−1/6 in the text should be written with parentheses, (√5−1)/6, to match the table layout and avoid ambiguity.
Circularity Check
No significant circularity; central claims are direct computations with self-contained protocol proofs.
full rationale
The paper's central claims — that |ϑ′⟩ (etc.) achieve S(ρ)=ln2, tangle 1, higher two-qubit entanglement than the W state, and enable perfect teleportation/superdense coding — are derived by direct computation from explicitly defined states. The states are given by amplitudes (e.g., |ϑ′⟩ = 1/2|001⟩ + 1/2|010⟩ + 1/√2|100⟩); ρ_A = diag(1/2,1/2) and τ_A(BC) = 1 follow from elementary partial traces and CKW formulas, so the properties are not inserted by definition of the measures. Tables 1–4 are 'straightforward calculations' from the Schmidt forms; the negativity formulas (9)–(11) are re-derived in the text from partial-transpose eigenvalues. The teleportation proof expands |φ⟩_a⊗|ξ⟩ into four mutually orthogonal measurement states with Bob's qubit carrying I, σ₃, σ₁, σ₁σ₃ images; the superdense-coding proof lists the four orthogonal encoded states explicitly. These proofs are self-contained even though [24] is cited. The vNEE formula (3) and monotonicity are cited to the author's [4,6], but the formula equals the standard binary entropy of the reduced density matrix of the SD form and is independently checkable; it is a cited tool, not a renamed input. The uniqueness theorems 1–3 rest on Lemmas 1–3 whose proofs are shown, plus 'a complicated calculation' for the maxima; an omitted proof is a rigor concern, not a circular reduction. A genuine correctness caveat (outside circularity scope): Eq. (3) is stated for the full W SLOCC class including λ₁ ≠ 0 in Eq. (1), but the formula is exact only when λ₁ = 0 (it omits the λ₁ contribution to the |1⟩⟨1| eigenvalue of ρ_A), which can affect the completeness of Lemmas 1–3; for the states actually tabulated (all λ₁ = 0) the numbers are correct. No fitted parameter is renamed as a prediction, and no equation reduces to another by construction.
Assumptions & free parameters
assumptions (4)
- standard math Every three-qubit state in the W SLOCC class has an LU-normal form λ0|000⟩ + λ1 e^{iφ}|100⟩ + λ2|101⟩ + λ3|110⟩ (Eq. (1)).
- standard math The 2-tangle, negativity, von Neumann entropy, and tangle τ are invariant under local unitary operations.
- domain assumption Formulas (3)–(4), (6), (9)–(11), (14)–(16) for vNEE, 2-tangles, negativities, and tangles in the W SLOCC class, cited from [4,6], are correct.
- domain assumption Alice and Bob can perform von Neumann measurements in arbitrary orthonormal bases and apply local Pauli unitaries in the teleportation/dense-coding protocols.
Cite this review
Pith. "Pith review of Comparing the three W-like states with the W state." pith.science (2026). https://pith.science/paper/ERSIVXTG
@misc{pith2026260713223,
author = {Pith},
title = {Pith review of: Comparing the three W-like states with the W state},
year = {2026},
howpublished = {\url{https://pith.science/paper/ERSIVXTG}},
note = {Machine review of arXiv:2607.13223}
}
abstract
In [Phy. Rev. Lett., 98, 260501 (2007)] and previous papers, the W-like states were used in many quantum communication schemes, distillation, teleportation and superdense coding. So far, no one has compared the W-like states with the W state. We investigate the properties of three W-like states named as $|\vartheta ^{\prime }\rangle $, $|\eta ^{\prime }\rangle $ and $|\xi ^{\prime }\rangle $. We point out that $|\vartheta ^{\prime }\rangle $ (resp. $|\eta ^{\prime }\rangle $ and $|\xi ^{\prime }\rangle $) has the maximal von Neumann entanglement entropy (vNEE) $S(\rho _{A})=\ln 2$ (resp. $S(\rho _{B})=\ln 2$ and $S(\rho _{C})=\ln 2$) and the maximal tangle $\tau _{A(BC)}=1$ (resp. $\tau _{B(AC)}=1$ and $\tau _{C(AB)}=1$) while the W state does not. We also indicate that if some one of three qubits is traced out, then the remaining two qubits of $|\vartheta ^{\prime }\rangle $ (resp. $|\eta ^{\prime }\rangle $ and $|\xi ^{\prime }\rangle $) are more entangled than the two qubits of the W state by several entanglement measures. It means that the three W-like states have higher robustness against some particle loss than the W state. We show that $|\vartheta ^{\prime }\rangle $, $|\eta ^{\prime }\rangle $ and $|\xi ^{\prime }\rangle $ can be suitable for perfect teleportation and superdense coding but the W state cannot.
Reference graph
Works this paper leans on
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[1]
(19) To guaranteeλ 0 ≥0,λ 1 in Eq
+ 1/4 = 0. (19) To guaranteeλ 0 ≥0,λ 1 in Eq. (19) must vanish. Then, from Eq. (19), we obtainλ 2 0 = 1/2. From λ2 0 +λ 2 2 +λ 2 3 = 1, we obtainλ 2 2 +λ 2 3 = 1/2. Thus, Eq. (1) reduces to Eq. (17). Conversely, for the state in Eq. (17), a calculation yieldsS(ρ A) = ln 2. Q.E.D. Whenλ 2 =λ 3 = 1/2 in Eq. (17), we obtain the state|ϑ⟩. Via Lemma 1, we next...
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[2]
(1) and via Eq
= 1/4.(18) Then, from P3 i=0 λ2 i = 1 in Eq. (1) and via Eq. (18), we obtain λ4 0 −λ 2 0(1−λ 2
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[3]
Thus,N AB = ρTA AB −1 /2 = 1 2 p 4λ2 3λ2 0 +λ 4 2 −λ 2 2)
By the definition of the negativity [5],N AB is equal to the sum of the abso- lute values of the negative eigenvalues of the partial transposeρ TA AB. Thus,N AB = ρTA AB −1 /2 = 1 2 p 4λ2 3λ2 0 +λ 4 2 −λ 2 2) . Similarly, we can derive NAC andN BC . Letτ x(yz) , wherex(yz) = A(BC), B(AC), C(AB), stand for the tangle [22], andAτ x(yz) for the average tangl...
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[4]
(3),S(ρ B) andS(ρ C) are the functions ofλ 2
From Eq. (3),S(ρ B) andS(ρ C) are the functions ofλ 2. The derivatives ofS(ρ B) andS(ρ C) with respect toλ 2 are S(ρC)′ =− 2λ2 −4λ 3 2p 1−4(λ 2 2 −λ 4 2) ln 1− p 1−4(λ 2 2 −λ 4 2) 1 + p 1−4(λ 2 2 −λ 4 2) , S(ρB)′ =−2λ 2 ln 1 + 2λ2 2 1−2λ 2 2 . Whenλ 2 = 1/2,S(ρ B)′ +S(ρ C)′ = 0. One can check thatS(ρ B) +S(ρ C) reaches its maximum at λ2 = 1/2. Thus, maxA ...
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[5]
From P3 i=0 λ2 i = 1 in Eq
= 1/4. From P3 i=0 λ2 i = 1 in Eq. (1), we obtain λ4 3 −λ 2 3(1−λ 2
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[6]
(23) Similarly, to guaranteeλ 3 ≥0,λ 1 in Eq
+ 1/4 = 0. (23) Similarly, to guaranteeλ 3 ≥0,λ 1 in Eq. (23) must vanish. Then, from Eq. (23) we obtainλ 2 3 = 1/2. Fromλ 2 0 +λ2 2 +λ2 3 = 1, we obtainλ 2 0 +λ2 2 = 1/2. Thus, Eq. (1) reduces to Eq. (22). Conversely, for the state in Eq. (22), a calculation yieldsS(ρ B) = ln 2. Q.E.D. Whenλ 0 =λ 2 = 1/2 in Eq. (22), we obtain the state|η⟩. Via Lemma 2, ...
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[7]
Via P3 i=0 λ2 i = 1 in Eq
= 1/4. Via P3 i=0 λ2 i = 1 in Eq. (1), we obtain λ4 2 −λ 2 2(1−λ 2
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[8]
(25) Similarly, to guaranteeλ 2 ≥0,λ 1 in Eq
+ 1/4 = 0. (25) Similarly, to guaranteeλ 2 ≥0,λ 1 in Eq. (25) must vanish. Then, from Eq. (25) we obtainλ 2 2 = 1/2. Fromλ 2 0 +λ2 2 +λ2 3 = 1, we obtainλ 2 0 +λ2 3 = 1/2. Thus, Eq. (1) reduces to Eq. (24). Conversely, for the state in Eq. (24), a calculation yieldsS(ρ C) = ln 2. Q.E.D. Whenλ 0 =λ 3 = 1/2 in Eq. (24), we obtain the state|ξ⟩. Via Lemma 3, ...
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Reviewed August 2, 2026 · model on record in the stance chip above.
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